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Computer Optics, 2019, Volume 43, Issue 4, Pages 611–617 (Mi co684)  

IMAGE PROCESSING, PATTERN RECOGNITION

Detection of the homotopy type of an object using differential invariants of an approximating map

S. V. Kurochkin

Russian Academy of Science, Federal Research Centre “Computer Science and Control”, Moscow, Russia

Abstract: A method of topological data analysis is proposed that allows one to find out the homotopy type of the object under study. Unlike mature and widely used methods based on persistent homologies, our method is based on computing differential invariants of some map associated with an approximating map. Differential topology tools and the analogy with the main result in Morse theory are used. The approximating map can be constructed in the usual way using a neural network or otherwise. The method allows one to identify the homotopy type of an object in the plane because the number of circles in the homotopy equivalent object representation as a wedge is expressed through the degree of some map associated with the approximating map. The performance of the algorithm is illustrated by examples from the MNIST database and transforms thereof. Generalizations and open questions relating to a higher-dimension case are discussed.

Keywords: machine learning, topological invariants, degree of a map, image processing.

DOI: https://doi.org/10.18287/2412-6179-2019-43-4-611-617

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Full text: http://www.computeroptics.smr.ru/.../430413.html
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Received: 19.11.2018
Accepted:04.07.2019

Citation: S. V. Kurochkin, “Detection of the homotopy type of an object using differential invariants of an approximating map”, Computer Optics, 43:4 (2019), 611–617

Citation in format AMSBIB
\Bibitem{Kur19}
\by S.~V.~Kurochkin
\paper Detection of the homotopy type of an object using differential invariants of an approximating map
\jour Computer Optics
\yr 2019
\vol 43
\issue 4
\pages 611--617
\mathnet{http://mi.mathnet.ru/co684}
\crossref{https://doi.org/10.18287/2412-6179-2019-43-4-611-617}


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